MAXIMUM A POSTERIORI SUPER RESOLUTION BASED ON SIMULTANEOUS NONSTATIONARY RESTORATION, INTERPOLATION

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MAXIMUM A POSTERIORI SUPER RESOLUTION BASED ON
SIMULTANEOUS NON-STATIONARY RESTORATION,
INTERPOLATION AND FAST REGISTRATIONby

• Giannis Chantas1, Nikolas P. Galatsanos1 and
  Nathan Woods2
• 1Department of Computer Science,
• University of Ioannina, Ioannina, Greece 45110
• 2Binary Machines, Inc, 1320 Tower Rd., 60173,
  Schaumburg, IL

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Outline

• Definition of Super Resolution Problem/Contributio
  n
• Imaging Model
• Image Prior Model
• Hierarchical Non Stationary Image Prior
• Super Resolution Algorithm (MAP)
• Image Reconstruction
• Fast Registration
• Experiments
• 6. Conclusions Future Work

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Definition of Super-resolution/Contributions

• Reconstruct high resolution image from multiple
  non registered degraded low resolution images
• Two contributions
• New non stationary image prior
• Fast Newton-Raphson based registration in DFT

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Imaging Model

• yi observation ( vector)
• x unknown high resolution image (vector
  ).
• S(di) Convolutional translation operator
  (Shannon Interpolator)
• Translation parameter di unknown
• Hi Convolutional blurring operator ( matrix),
  assumed known
• D decimation operator, N x dN matrix, d
  decimation factor
• ni vector, AWG noise unknown
• OBJECTIVE Reconstruction of unknown image x

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Ill Posedness of Reconstruction

• Vector notation
• Reconstruction only from observations ill-posed
• Solution regularize the estimate by
  incorporating prior knowledge
• Prior knowledge in MAP introduction of image
  prior

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Image Prior Model Based on Prediction

• Prediction based on spatial smoothness of images
• SAR prediction model
• prediction residual at image location
• Random variable
• Matrix vector form
• Laplacian operator, matrix

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Stationary Model

• Statistics of residual do not vary with
  spatial location
• Advantage few parameters to estimate, easy to
  solve using ML (E-M)
• Disadvantage cannot model image edge structure

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Non-Stationary Image Model

• Residuals of the first order differences in four
  directions
• Residuals assumed spatially varying

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Non-Stationary Image Model

• Induced image prior
• Advantage Can capture image edge structure
• Disadvantage Too many parameters to estimate!!

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Solution to Over-parameterization Problem

• Problem 4NH parameters ( ) estimated from PN
  data points
• Solution
• Assume random variable
• Introduction of conjugate hyper prior (Gamma)

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MAP Approach

• MAP approach steps
• Posterior obtained from Bayes Rule
• Estimate variables and shift parameters by
  maximizing posterior

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MAP Approach

• MAP estimate minimization of negative logarithm
  of posterior

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MAP Algorithm

• Iterative scheme set gradients
  
• alternatively equal to zero

• Linear system solved by Conjugate-Gradients
  algorithm

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MAP Algorithm - Fast Registration

• In closed form impossible to calculate
• Solution resort to minimization (equivalent
  to Registration)
• Newton-Raphson algorithm for minimization

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MAP Algorithm - Fast Registration

• Analytical first and second derivatives
  calculation of the norm in DFT
• Convenient form in the DFT domain
• Hi and S circulant, diagonal in DFT
• D sparse in DFT
• Exponential form of diagonal elements of S in
  DFT analytical derivatives
• Quadratic convergence of Newton-Raphson algorithm

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Experiments
Acknowledgment S. Farsiu and P. Milanfar, EE
Department of University of California Santa Cruz
provided the low resolution used in this work

• 20 non registered low resolution images of size
  64x64
• Target images size 128x128

4 of 20 observations
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Experiments
MAP non stationary 2x super-resolved image.
Stationary 2x super-resolved image.
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Experiments

• 4 non registered low resolution images of size
  128x128
• Target images size 512x512

Low resolution observations
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Experiments
Stationary 4x super-resolved image.
MAP non stationary 4x super-resolved image.
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Conclusions and Future Work

• Non stationary model yields visually more
  pleasing results (less ringing artifact)
• Registration faster than previous works
• Future Work
• New Image prior Model
• PSF estimation